Factors and Products
Date:
Math 10 Plus Notes
Background Information: Factors and Prime Numbers
Factor
Factors are numbers that you _________________________ together
to get another number. The factors of 6 are _______________ and
________________. Numbers can have many factors.
Example
List the factors of 36.
Prime
Number
A prime number is a number whose ___________ factors are
__________ and itself. ________ is an example of a prime number.
Composite
Number
A composite number is a number that has factors _______________
___________________ one and itself. Any number that is ___________ a
prime number is a composite number.
Prime
Factorization
A prime factorization is when a number is represented as a
product of its prime factors. To give a prime factorization, you
must find which prime numbers make up the original number.
Finding Prime Factors
Method 1: Factor Trees
Find two factors of the number then find two factors for each
of the factors. Continue this process until all factors are prime.
Example
72
216
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Factors and Products
Math 10 Plus Notes
Practice
Date:
Find the prime factors using factor trees:
a) 45
b) 88
c) 108
Method 2: Repeated Division
Begin by dividing by the smallest prime number (ie. 2).
Continue to divide by 2 until it is no longer a factor (the result
is not a whole number). Then divide by the next largest prime
number for as long as gives a whole number. Continue this
process until the result is 1.
Example
Practice
60
75
Find the prime factors using repeated division:
a) 88
b) 56
c) 117
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Factors and Products
Math 10 Plus Notes
Think
Date:
Can you create a different factor tree then the one given?
100
/
\
10
10
/
\
/ \
5
2 5
2
Think
Two numbers have the prime factorizations given below.
What can you conclude about the two numbers?
#1: 2x2x3x5x7
#2: 7x5x2x3x2
Note
When a number’s prime factorization is large, it is often
helpful to re-write the factors as ____________________________
instead of sets of factors. An exponent shows us the number of
times a number is ____________________________________ by itself.
For example, 24 = 2 x 2 x 2 x 2.
Example
Find the prime factorization of 432 then re-write in exponent
form.
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Factors and Products
Math 10 Plus Notes
Practice
Date:
Find the prime factorization and re-write in exponent form for
the numbers given below:
a) 320
b) 576
c) 1260
Think
Why do 0 and 1 have no prime factors?
Think
How many sets of prime factors does each number have?
Activity
Find a partner. Take turns drawing two cards from the deck to
create a 2 digit number. Find the prime factors of the number
and then add them up. This is your score. The first person to
reach 50 points is the winner.
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Factors and Products
Date:
Math 10 Plus Notes
Greatest Common Factor and Least Common Multiples
Greatest
Common
Factor
The greatest common factor (GCF) of two or more numbers is
the ______________________ factor that the numbers ________________.
For example, the prime factors of 14 are 7 and 2 and the prime
factors of 21 are 7 and 3. The greatest common factor of 14
and 21 is ________.
Method 1: Rainbow
Use division to determine all of the factors of each number and
record as a rainbow. List the common factors from each
rainbow and choose the largest.
Example
Find the GCF of 198 and 138.
Practice
Find the GCF of the numbers given below:
a) 12 and 18
b) 170 and 220
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Factors and Products
Math 10 Plus Notes
Date:
Method 2: List and circle the factors
List the factors for each of the numbers and circle all the
factors which are common. Identify the largest.
Example
Find the GCF of 36 and 48.
Practice
Find the GCF of 210 and 175.
Method 3: Using Factor Trees
Write the prime factorization of each number and multiply
together all prime factors that appear in both factorizations.
Example
Find the GCF of 180, 216 and 252.
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Factors and Products
Math 10 Plus Notes
Practice
Date:
Find the GCF of the three numbers below using factor trees:
a) 42, 90, 120
Multiple
A multiple is the result of ________________________ a number by an
integer (a positive or negative whole number). 6, 9, 12, 15 and
18 are all multiples of __________ (3x2 = 6, 3x3 = 9, 3x4 = 12…).
Least
Common
Multiple
The least common multiple (LCM) of two or more numbers is
the ____________________________ number that is a multiple of each.
Method 1: Lists
List the multiples of each number until the same multiple
appears in each list.
Example
Find the LCM of 9, 12 and 15.
Practice
Find the LCM of 10 and 12.
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Factors and Products
Math 10 Plus Notes
Date:
Method 2: Cross Check
List the multiples of one of the numbers and then divide the
multiples by the other numbers. The first multiple that is
divisible by the other numbers is the LCM.
Example
Use the cross check method to find the LCM of 12 and 14.
Practie
Use the cross check method to find the LCM of 12, 18, and 24.
Method 3: Prime Factorization
Find the prime factorization of each number. The LCM is the
product of each prime factor raised to its greatest power.
Example
Use prime factors to find the LCM of 18 and 24.
Practice
Use prime factors to find the LCM of 36 and 52.
NOTE
You do not need to know how to use multiple methods. You
may choose the method you prefer for finding GCF and LCM.
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Factors and Products
Math 10 Plus Notes
Date:
Using LCM and GCF to Simplify Fractions
We can use the GCF to simplify fractions. If the GCF of the
numerator and denominator is known, it can be cancelled out
to leave a fraction in simplest terms.
Example
Use the GCF of 325 and 400 to simplify the fraction below:
325 =
450
Practice
Use the GCF of 210 and 120 to simplify the fraction below:
120 =
210
The LCM can be used to find a common denominator for two
fractions, which is necessary when adding and subtracting
fractions.
Example
Use the LCM to add the fractions below:
9 +7 =
25 15
Practice
Use the LCM to subtract the fractions below:
10 – 4 =
5
9
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Factors and Products
Math 10 Plus Notes
Date:
Using LCM and GCF to Solve Word Problems
Both the LCM and the GCF can be used to solve real-world
problems, like the exams below:
Example
A fruit basket contains apples and oranges. There are 10
apples and 15 oranges. If each basket is the same, what is the
largest number of fruit baskets that can be made? How many
apples and oranges will be in each basket?
We need to find the ___________ to solve this problem.
Example
There is a stack of tiles with a length of 84 cm and a width of
63 cm. We want to arrange the tiles in a perfect square. What
is the smallest side length the square can have?
We need to find the ____________ to solve this problem.
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Factors and Products
Math 10 Plus Notes
Using LCM
Date:
Use the LCM to solve a problem when the solution is a larger
number than the numbers given in the problem. For example:
a) To use smaller items to create a larger object (like the tile
example on the previous page).
b) To get multiples of two or more different things in order to
have enough.
c) To figure out when two or more events will happen again at
the same time.
d) To figure out two or more patterns will repeat at the same
time.
A key indication that LCM is needed is when you are asked to
find the smallest possible number.
Practice
Amna exercises every 3 days. Jamal exercises every 2 days.
They both exercised today. In how many days will they
exercise on the same day again?
Practice
How many times will they exercise together in one month (30
days) if they exercise at the same time on the first day of the
month?
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Factors and Products
Math 10 Plus Notes
Using GCF
Date:
Use the GCF to solve a problem when the solution is smaller
than the numbers given in the problem. For example:
a) To divide things into smaller sections (like the fruit basket
example on the previous page).
b) To arrange two or more different types of things into
groups.
A key indication that GCF is needed is when you are asked to
find the largest possible number.
Practice
There are 16 boys and 12 girls in a class. What is the largest
number of groups that can be made with an equal number of
boys and girls? How many boys and how many girls are in
each group?
Practice
Omar has 20 five dirham bills, 18 ten dirham bills and 24
twenty dirham bills. What is the largest number of groups he
can divide his money into if each group must contain one of
each bill?
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Factors and Products
Math 10 Plus Notes
Date:
Perfect Squares, Perfect Cubes and Their Roots
Perfect
Square
A perfect square has a square root that is a ____________ number
(a square root is a number that is _____________________________ by
______________ to give the square number). For example, 25 is a
perfect square because _____ x ______ = 25. The square root of
25 is ________.
We can use factor trees in order to decide if a number is
perfect square and to find its square root. If we can divide the
prime factors into __________ equal groups, then the number is a
perfect square because there is a whole number that squared
results in the number. The square root is the product of
___________ of the groups of factors.
Example
Use factor trees to show that 64 is a perfect square and that 8
is its square root.
Practice
Use a factor tree to decide if 324 is a perfect square. If so, what
is its square root?
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Factors and Products
Date:
Math 10 Plus Notes
Practice
Use a factor tree to decide if 288 is a perfect square. If so, find
its square root. Do the same for 441.
Perfect
Cube
A perfect cube has a cube root that is a whole number. In other
words, a whole number is multiplied by itself ____________ times
to produce a perfect cube.
Factor trees can be used in the same to determine if a number
is a perfect cube and to find its cube root. In this case, a
number is a perfect cube if the prime factors can be divided
into three _________________ groups. The cube root is the product
of ______________ of those groups.
Example
Use factor trees to show that 216 is a perfect cube and to show
that its cube root is 6.
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Factors and Products
Date:
Math 10 Plus Notes
Practice
Use a factor tree to decide if 64 is a perfect cube. If so, find its
square root. Do the same for 1728.
Note
A number that is both a perfect square and a perfect cube has
prime factors that can be divided into both ______ equal groups
and ________ equal groups.
Example
Use factor trees to show that 729 is both a perfect square and
a perfect cube.
Practice
Use factor trees to decide if 512 is a perfect square, a perfect
cube, both or neither.
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Factors and Products
Math 10 Plus Notes
Date:
Example
What is the length of a square piece of land that has an area of
1764 m2? Use prime factors.
Example
The volume of a cube is 125 cm3. What is the side length of the
cube? Use prime factors.
Think
The lowest whole number that is both a perfect square and a
perfect cube is 1. What is the next smallest number that is a
perfect square and perfect cube?
Challenge
Find the 5th root of 1025 using prime factors.
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